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Let be a regular, algebraic, essentially self-dual cuspidal automorphic representation of , where is a totally real field and is at most . We show that for all primes , the -adic Galois representations associated to are irreducible, and for all but finitely many primes , the mod Galois representations associated to are also irreducible. We also show that the Lie algebras of the Zariski closures of the -adic representations are independent of .
We prove the compatibility of the local and global Langlands correspondences at places dividing for the -adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of over an imaginary CM field, under the assumption that the automorphic representations have Iwahori-fixed vectors at places dividing and have Shin-regular weight.
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